Theorem kur14 31198

Description: Kuratowski's closure-complement theorem. There are at most 14 sets which can be obtained by the application of the closure and complement operations to a set in a topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)

Hypotheses

Ref	Expression
kur14.x	⊢ 𝑋 = ∪ 𝐽
kur14.k	⊢ 𝐾 = (cls‘𝐽)
kur14.s	⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}

Assertion

Ref	Expression
kur14	⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝑆 ∈ Fin ∧ (#‘𝑆) ≤ ;14))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐽,𝑦   𝑥,𝑋

Allowed substitution hints:   𝑆(𝑥,𝑦)   𝐾(𝑥,𝑦)   𝑋(𝑦)

Proof of Theorem kur14

Step	Hyp	Ref	Expression
1		kur14.s	. . . . . 6 ⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}
2		eleq1 2689	. . . . . . . . 9 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → (𝐴 ∈ 𝑥 ↔ if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥))
3	2	anbi1d 741	. . . . . . . 8 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → ((𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) ↔ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)))
4	3	rabbidv 3189	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} = {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)})
5	4	inteqd 4480	. . . . . 6 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)})
6	1, 5	syl5eq 2668	. . . . 5 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)})
7	6	eleq1d 2686	. . . 4 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → (𝑆 ∈ Fin ↔ ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ∈ Fin))
8	6	fveq2d 6195	. . . . 5 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → (#‘𝑆) = (#‘∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}))
9	8	breq1d 4663	. . . 4 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → ((#‘𝑆) ≤ ;14 ↔ (#‘∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}) ≤ ;14))
10	7, 9	anbi12d 747	. . 3 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → ((𝑆 ∈ Fin ∧ (#‘𝑆) ≤ ;14) ↔ (∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ∈ Fin ∧ (#‘∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}) ≤ ;14)))
11		kur14.x	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
12		unieq 4444	. . . . . . . . . 10 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ∪ 𝐽 = ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
13	11, 12	syl5eq 2668	. . . . . . . . 9 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝑋 = ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
14	13	pweqd 4163	. . . . . . . 8 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝑋 = 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
15	14	pweqd 4163	. . . . . . 7 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝒫 𝑋 = 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
16	13	sseq2d 3633	. . . . . . . . . . 11 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴 ⊆ 𝑋 ↔ 𝐴 ⊆ ∪ if(𝐽 ∈ Top, 𝐽, {∅})))
17		sn0top 20803	. . . . . . . . . . . . . 14 ⊢ {∅} ∈ Top
18	17	elimel 4150	. . . . . . . . . . . . 13 ⊢ if(𝐽 ∈ Top, 𝐽, {∅}) ∈ Top
19		uniexg 6955	. . . . . . . . . . . . 13 ⊢ (if(𝐽 ∈ Top, 𝐽, {∅}) ∈ Top → ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∈ V)
20	18, 19	ax-mp 5	. . . . . . . . . . . 12 ⊢ ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∈ V
21	20	elpw2 4828	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ↔ 𝐴 ⊆ ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
22	16, 21	syl6bbr 278	. . . . . . . . . 10 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴 ⊆ 𝑋 ↔ 𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅})))
23	22	ifbid 4108	. . . . . . . . 9 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → if(𝐴 ⊆ 𝑋, 𝐴, ∅) = if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅))
24	23	eleq1d 2686	. . . . . . . 8 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ↔ if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥))
25	13	difeq1d 3727	. . . . . . . . . . 11 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝑋 ∖ 𝑦) = (∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦))
26		kur14.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (cls‘𝐽)
27		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (cls‘𝐽) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
28	26, 27	syl5eq 2668	. . . . . . . . . . . 12 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝐾 = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
29	28	fveq1d 6193	. . . . . . . . . . 11 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐾‘𝑦) = ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦))
30	25, 29	preq12d 4276	. . . . . . . . . 10 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} = {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)})
31	30	sseq1d 3632	. . . . . . . . 9 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ({(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥 ↔ {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥))
32	31	ralbidv 2986	. . . . . . . 8 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥))
33	24, 32	anbi12d 747	. . . . . . 7 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ((if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) ↔ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)))
34	15, 33	rabeqbidv 3195	. . . . . 6 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} = {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)})
35	34	inteqd 4480	. . . . 5 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} = ∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)})
36	35	eleq1d 2686	. . . 4 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ∈ Fin ↔ ∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)} ∈ Fin))
37	35	fveq2d 6195	. . . . 5 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (#‘∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}) = (#‘∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)}))
38	37	breq1d 4663	. . . 4 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ((#‘∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}) ≤ ;14 ↔ (#‘∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)}) ≤ ;14))
39	36, 38	anbi12d 747	. . 3 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ((∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ∈ Fin ∧ (#‘∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}) ≤ ;14) ↔ (∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)} ∈ Fin ∧ (#‘∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)}) ≤ ;14)))
40		eqid 2622	. . . 4 ⊢ ∪ if(𝐽 ∈ Top, 𝐽, {∅}) = ∪ if(𝐽 ∈ Top, 𝐽, {∅})
41		eqid 2622	. . . 4 ⊢ (cls‘if(𝐽 ∈ Top, 𝐽, {∅})) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅}))
42		eqid 2622	. . . 4 ⊢ ∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)} = ∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)}
43		0elpw 4834	. . . . . 6 ⊢ ∅ ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅})
44	43	elimel 4150	. . . . 5 ⊢ if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅})
45		elpwi 4168	. . . . 5 ⊢ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) → if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ⊆ ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
46	44, 45	ax-mp 5	. . . 4 ⊢ if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ⊆ ∪ if(𝐽 ∈ Top, 𝐽, {∅})
47	18, 40, 41, 42, 46	kur14lem10 31197	. . 3 ⊢ (∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)} ∈ Fin ∧ (#‘∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)}) ≤ ;14)
48	10, 39, 47	dedth2h 4140	. 2 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐽 ∈ Top) → (𝑆 ∈ Fin ∧ (#‘𝑆) ≤ ;14))
49	48	ancoms 469	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝑆 ∈ Fin ∧ (#‘𝑆) ≤ ;14))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  ifcif 4086  𝒫 cpw 4158  {csn 4177  {cpr 4179  ∪ cuni 4436  ∩ cint 4475   class class class wbr 4653  ‘cfv 5888  Fincfn 7955  1c1 9937   ≤ cle 10075  4c4 11072  ;cdc 11493  #chash 13117  Topctop 20698  clsccl 20822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-hash 13118  df-top 20699  df-topon 20716  df-cld 20823  df-ntr 20824  df-cls 20825

This theorem is referenced by: (None)